Filtering problem (stochastic processes)

In the theory of stochastic processes, the filtering problem is a mathematical model for a number of filtering problems in signal processing and the like. The general idea is to form some kind of "best estimate" for the true value of some system, given only some (potentially noisy) observations of that system. The problem of optimal non-linear filtering (even for the non-stationary case) was solved by Ruslan L. Stratonovich (1959,^[1] 1960^[2]), see also Harold J. Kushner's work ^[3] and Moshe Zakai's, who introduced a simplified dynamics for the unnormalized conditional law of the filter^[4] known as Zakai equation. The solution, however, is infinite-dimensional in the general case.^[5] Certain approximations and special cases are well-understood: for example, the linear filters are optimal for Gaussian random variables, and are known as the Wiener filter and the Kalman-Bucy filter. More generally, as the solution is infinite dimensional, it requires finite dimensional approximations to be implemented in a computer with finite memory. A finite dimensional approximated nonlinear filter may be more based on heuristics, such as the Extended Kalman Filter or the Assumed Density Filters,^[6] or more methodologically oriented such as for example the Projection Filters,^[7] some sub-families of which are shown to coincide with the Assumed Density Filters.^[8]

In general, if the separation principle applies, then filtering also arises as part of the solution of an optimal control problem. For example, the Kalman filter is the estimation part of the optimal control solution to the Linear-quadratic-Gaussian control problem.

The mathematical formalism

Consider a probability space (Ω, Σ, P) and suppose that the (random) state Y_t in n-dimensional Euclidean space Rⁿ of a system of interest at time t is a random variable Y_t : Ω → Rⁿ given by the solution to an Itō stochastic differential equation of the form

$\mathrm{d} Y_{t} = b(t, Y_{t}) \, \mathrm{d} t + \sigma (t, Y_{t}) \, \mathrm{d} B_{t},$

where B denotes standard p-dimensional Brownian motion, b : [0, +∞) × Rⁿ → Rⁿ is the drift field, and σ : [0, +∞) × Rⁿ → R^n×p is the diffusion field. It is assumed that observations H_t in R^m (note that m and n may, in general, be unequal) are taken for each time t according to

$H_{t} = c(t, Y_{t}) + \gamma (t, Y_{t}) \cdot \mbox{noise}.$

Adopting the Itō interpretation of the stochastic differential and setting

$Z_{t} = \int_{0}^{t} H_{s} \, \mathrm{d} s,$

this gives the following stochastic integral representation for the observations Z_t:

$\mathrm{d} Z_{t} = c(t, Y_{t}) \, \mathrm{d} t + \gamma (t, Y_{t}) \, \mathrm{d} W_{t},$

where W denotes standard r-dimensional Brownian motion, independent of B and the initial condition X₀, and c : [0, +∞) × Rⁿ → Rⁿ and γ : [0, +∞) × Rⁿ → R^n×r satisfy

$\big| c (t, x) \big| + \big| \gamma (t, x) \big| \leq C \big( 1 + | x | \big)$

for all t and x and some constant C.

The filtering problem is the following: given observations Z_s for 0 ≤ s ≤ t, what is the best estimate Ŷ_t of the true state Y_t of the system based on those observations?

By "based on those observations" it is meant that Ŷ_t is measurable with respect to the σ-algebra G_t generated by the observations Z_s, 0 ≤ s ≤ t. Denote by K = K(Z, t) be collection of all Rⁿ-valued random variables Y that are square-integrable and G_t-measurable:

$K = K(Z, t) = L^{2} (\Omega, G_{t}, \mathbf{P}; \mathbf{R}^{n}).$

By "best estimate", it is meant that Ŷ_t minimizes the mean-square distance between Y_t and all candidates in K:

$\mathbf{E} \left[ \big| Y_{t} - \hat{Y}_{t} \big|^{2} \right] = \inf_{Y \in K} \mathbf{E} \left[ \big| Y_{t} - \hat{Y} \big|^{2} \right]. \qquad \mbox{(M)}$

Basic result: orthogonal projection

The space K(Z, t) of candidates is a Hilbert space, and the general theory of Hilbert spaces implies that the solution Ŷ_t of the minimization problem (M) is given by

$\hat{Y}_{t} = P_{K(Z, t)} \big( X_{t} \big),$

where P_K(Z,t) denotes the orthogonal projection of L²(Ω, Σ, P; Rⁿ) onto the linear subspace K(Z, t) = L²(Ω, G_t, P; Rⁿ). Furthermore, it is a general fact about conditional expectations that if F is any sub-σ-algebra of Σ then the orthogonal projection

$P_{K} : L^{2} (\Omega, \Sigma, \mathbf{P}; \mathbf{R}^{n}) \to L^{2} (\Omega, F, \mathbf{P}; \mathbf{R}^{n})$

is exactly the conditional expectation operator E[·|F], i.e.,

$P_{K} (X) = \mathbf{E} \big[ X \big | F \big].$

Hence,

$\hat{Y}_{t} = P_{K(Z, t)} \big( X_{t} \big) = \mathbf{E} \big[ X_{t} \big | G_{t} \big].$

This elementary result is the basis for the general Fujisaki-Kallianpur-Kunita equation of filtering theory.

References

• Jazwinski, Andrew H. (1970). Stochastic Processes and Filtering Theory. New York: Academic Press. ISBN 0-12-381550-9.
• Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1. (See Section 6.1)

1. Stratonovich, R. L. (1959). Optimum nonlinear systems which bring about a separation of a signal with constant parameters from noise. Radiofizika, 2:6, pp. 892-901.
2. Stratonovich, R.L. (1960). Application of the Markov processes theory to optimal filtering. Radio Engineering and Electronic Physics, 5:11, pp.1-19.
3. Kushner, Harold. (1967). Nonlinear filtering: The exact dynamical equations satisfied by the conditional mode. Automatic Control, IEEE Transactions on Volume 12, Issue 3, Jun 1967 Page(s): 262 - 267
4. Zakai, Moshe (1969), On the optimal filtering of diffusion processes. Zeit. Wahrsch. 11 230–243. MR 242552, Zbl 0164.19201, doi:10.1007/BF00536382
5. Mireille Chaleyat-Maurel and Dominique Michel. Des resultats de non existence de filtre de dimension finie. Stochastics, 13(1+2):83-102, 1984.
6. Maybeck, Peter S., Stochastic models, estimation, and control, Volume 141, Series Mathematics in Science and Engineering, 1979, Academic Press
7. Damiano Brigo, Bernard Hanzon and François LeGland, A Differential Geometric approach to nonlinear filtering: the Projection Filter, I.E.E.E. Transactions on Automatic Control Vol. 43, 2 (1998), pp 247--252.
8. Damiano Brigo, Bernard Hanzon and François Le Gland, Approximate Nonlinear Filtering by Projection on Exponential Manifolds of Densities, Bernoulli, Vol. 5, N. 3 (1999), pp. 495--534